The quadratic equation x^2-3x+2=0 has the "correct" number of solutions modulo 5 and 7. However, modulo 6 the equation has four solutions; namely, 1, 2, 4, and 5. For what positive integers n does the equation x^2-3x+2=0 have exactly two incongruent solutions modulo n?

(In reply to re(4): Proof by Richard)

Please forgive the following pickiness.  You said more than Tristan about incongruence but I'm not sure everyone follows your argument.

You wrote "Hence ac and bd are an x-1 and an x-2 in some order for an x that is neither 1 nor 2 mod n=ab", without saying why x is neither 1 nor 2 mod n=ab.  Part of the why is in your earlier statement that c is relatively prime to b.  Novices might have some trouble filling in the details.

Posted by McWorter on 2005-09-04 03:51:18